Angle A of rhombus ABCD measures 120°. If one side of the rhombus is 10 units, what is the length of the longer diagonal?

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Accepted Answer

The diagonals of a rhombus divide the rhombus into 4 congruent right triangles. Since angle A measures 120°, angle D also measures 120° and angles B and C each measure 60°. Each right triangle is made up of a right angle, a 30° angle (the angle formed by the diagonal bisecting the 60° angle of the rhombus), and a 60° angle (the angle formed by the diagonal bisecting the 120° angle of the rhombus). The bases of each triangle are made up of half the shorter diagonal (opposite the 30° angle) and half the longer diagonal (opposite the 60° angle). The hypotenuse is the side of the rhombus, 10 units. Since these right triangles are 30-60-90 right triangles, the measure of the shorter base is 10 ⁄ 2 = 5 units, and the measure of the longer base is 5√3 units. The longer base is half the length of the longer diagonal; therefore, the length of the longer diagonal is 2(5√3) = 10√3 units.

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